Metamath Proof Explorer

Theorem mapdpglem3

Description: Lemma for mapdpg . Baer p. 45, line 3: "infer ... the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope $d g w z ph locally to avoid clashes with later substitutions into ph .) (Contributed by NM, 18-Mar-2015)

Ref Expression
Hypotheses mapdpglem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
mapdpglem.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
mapdpglem.s ⊢ − = ( -g ‘ 𝑈 )
mapdpglem.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
mapdpglem.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
mapdpglem.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
mapdpglem.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
mapdpglem1.p ⊢ ⊕ = ( LSSum ‘ 𝐶 )
mapdpglem2.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
mapdpglem3.f ⊢ 𝐹 = ( Base ‘ 𝐶 )
mapdpglem3.te ⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
mapdpglem3.a ⊢ 𝐴 = ( Scalar ‘ 𝑈 )
mapdpglem3.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
mapdpglem3.t ⊢ · = ( ·𝑠 ‘ 𝐶 )
mapdpglem3.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
mapdpglem3.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
mapdpglem3.e ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
Assertion mapdpglem3 ( 𝜑 → ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )

Proof

Step Hyp Ref Expression
1 mapdpglem.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 mapdpglem.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3 mapdpglem.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 mapdpglem.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 mapdpglem.s ⊢ − = ( -g ‘ 𝑈 )
6 mapdpglem.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7 mapdpglem.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 mapdpglem.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 mapdpglem.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10 mapdpglem.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
11 mapdpglem1.p ⊢ ⊕ = ( LSSum ‘ 𝐶 )
12 mapdpglem2.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
13 mapdpglem3.f ⊢ 𝐹 = ( Base ‘ 𝐶 )
14 mapdpglem3.te ⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
15 mapdpglem3.a ⊢ 𝐴 = ( Scalar ‘ 𝑈 )
16 mapdpglem3.b ⊢ 𝐵 = ( Base ‘ 𝐴 )
17 mapdpglem3.t ⊢ · = ( ·𝑠 ‘ 𝐶 )
18 mapdpglem3.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
19 mapdpglem3.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
20 mapdpglem3.e ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
21 20 oveq1d ⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
22 14 21 eleqtrd ⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
23 r19.41v ⊢ ( ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
24 1 7 8 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
25 eqid ⊢ ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 )
26 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) )
27 25 26 13 17 12 ellspsn ⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) )
28 24 19 27 syl2anc ⊢ ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) )
29 1 3 15 16 7 25 26 8 lcdsbase ⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) = 𝐵 )
30 29 rexeqdv ⊢ ( 𝜑 → ( ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ↔ ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) )
31 28 30 bitrd ⊢ ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) )
32 31 anbi1d ⊢ ( 𝜑 → ( ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
33 23 32 bitr4id ⊢ ( 𝜑 → ( ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
34 33 exbidv ⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
35 df-rex ⊢ ( ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
36 34 35 bitr4di ⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
37 eqid ⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
38 37 lsssssubg ⊢ ( 𝐶 ∈ LMod → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) )
39 24 38 syl ⊢ ( 𝜑 → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) )
40 13 37 12 lspsncl ⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) )
41 24 19 40 syl2anc ⊢ ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) )
42 39 41 sseldd ⊢ ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( SubGrp ‘ 𝐶 ) )
43 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
44 1 3 8 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
45 4 43 6 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
46 44 10 45 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
47 1 2 3 43 7 37 8 46 mapdcl2 ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
48 39 47 sseldd ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( SubGrp ‘ 𝐶 ) )
49 18 11 42 48 lsmelvalm ⊢ ( 𝜑 → ( 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
50 36 49 bitr4d ⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) )
51 22 50 mpbird ⊢ ( 𝜑 → ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
52 ovex ⊢ ( 𝑔 · 𝐺 ) ∈ V
53 oveq1 ⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑤 𝑅 𝑧 ) = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
54 53 eqeq2d ⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) )
55 54 rexbidv ⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) )
56 52 55 ceqsexv ⊢ ( ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
57 56 rexbii ⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
58 rexcom4 ⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
59 57 58 bitr3i ⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ↔ ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
60 51 59 sylibr ⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )